Smallness in the context of universes

If Grothendieck universes are being used, then for UU a fixed Grothendieck universe, a category CC is UU-small if its collection of objects and collection of morphisms are both elements of UU. CC is essentially UU-small if there is a bijection from its set of morphisms to an element of UU (the same for the set of objects follows); this condition is non-evil.